Answer: I have a great respect for whomever accepted that responsibility.
Explanation: The word "whomever" is an object pronoun, that is to say that it belongs to the same type of pronouns than "her", "him" and "them". In contrast, "whoever" is a subject pronoun. "Whomever" can play the functions of direct object or object of the preposition in a sentence, In the sentence provided, the blank is preceded by a preposition ("for") and the element in the gap is functioning as an object; therefore, the word that fills in the blank is an object pronoun: whomever.
